×
Coco, Armando; Ekström, Sven-Erik; Russo, Giovanni; Serra-Capizzano, Stefano; Stissi, Santina Chiara

Spectral and norm estimates for matrix-sequences arising from a finite difference approximation of elliptic operators. (English) Zbl 1512.65242

Linear Algebra Appl. 667, 10-43 (2023).
The authors consider matrix-sequences arising from elliptic differential equations with Dirichlet boundary condition where the Laplacian and the boundary condition are approximated by using ad hoc finite differences. They provide spectral and norm estimates for these matrix-sequences based on several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix-sequences. Numerical experiments are conducted to confirm the correctness of their theoretical findings.
Reviewer: Weizhong Dai (Ruston)

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
15B05 Toeplitz, Cauchy, and related matrices
35J25 Boundary value problems for second-order elliptic equations

Keywords:

generating function; spectral symbol; approximation of differential operators
Cite Review PDF
Full Text: DOI arXiv
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Barbarino, G., A systematic approach to reduced GLT, BIT, 62, 681-743 (2022) · Zbl 1495.15035
[2] Barbarino, G.; Serra-Capizzano, S., Non-Hermitian perturbations of Hermitian matrix-sequences and applications to the spectral analysis of the numerical approximation of partial differential equations, Numer. Linear Algebra Appl., 27, 3, Article e2286 pp. (2020) · Zbl 1474.65454
[3] Bhatia, R., Matrix Analysis, Graduate Texts in Mathematics (Nov. 1996), Springer: Springer New York, NY · Zbl 0863.15001
[4] Chan, R. H.-F.; Jin, X.-Q., An Introduction to Iterative Toeplitz Solvers (2007), SIAM · Zbl 1146.65028
[5] Chertock, A.; Coco, A.; Kurganov, A.; Russo, G., A second-order finite-difference method for compressible fluids in domains with moving boundaries, Commun. Comput. Phys., 23, 230-263 (2018) · Zbl 1488.76097
[6] Coco, A.; Currenti, G.; Del Negro, C.; Russo, G., A second order finite-difference ghost-point method for elasticity problems on unbounded domains with applications to volcanology, Commun. Comput. Phys., 16, 4, 983-1009 (2014) · Zbl 1373.74097
[7] Coco, A.; Russo, G., Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains, J. Comput. Phys., 241, 464-501 (2013) · Zbl 1349.65555
[8] Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152, 2, 457-492 (1999) · Zbl 0957.76052
[9] Garoni, C.; Serra-Capizzano, S., Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 1 (2017), Springer · Zbl 1376.15002
[10] Garoni, C.; Serra-Capizzano, S., Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 2 (2018), Springer · Zbl 1448.47004
[11] Gibou, F.; Fedkiw, R. P., A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202, 2, 577-601 (2005) · Zbl 1061.65079
[12] Gibou, F.; Fedkiw, R. P.; Cheng, L.-T.; Kang, M., A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176, 1, 205-227 (2002) · Zbl 0996.65108
[13] Golinskii, L.; Serra-Capizzano, S., The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences, J. Approx. Theory, 144, 1, 84-102 (2007) · Zbl 1111.15013
[14] LeVeque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 4, 1019-1044 (1994) · Zbl 0811.65083
[15] Meurant, G., A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13, 3, 707-728 (1992) · Zbl 0754.65029
[16] Ng, M. K., Iterative Methods for Toeplitz Systems, Numerical Mathematics and Scientific Computation (2004) · Zbl 1059.65031
[17] Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 3, 220-252 (1977) · Zbl 0403.76100
[18] Serra-Capizzano, S., Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl., 366, 371-402 (2003) · Zbl 1028.65109
[19] Serra-Capizzano, S., The GLT class as a generalized Fourier analysis and applications, Linear Algebra Appl., 419, 1, 180-233 (2006) · Zbl 1109.65032
[20] Serra-Capizzano, S.; Tilli, P., On unitarily invariant norms of matrix-valued linear positive operators, J. Inequal. Appl., 7, 3, 309-330 (2002) · Zbl 1055.15039
[21] Tyrtyshnikov, E.; Zamarashkin, N., Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships, Linear Algebra Appl., 270, 1-3, 15-27 (1998) · Zbl 0890.15006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.